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Conformal Field Theory
Introduction Mapping functions A "mapping function" is a mathematical function that operates on a multi-dimensional "input set" (equivalent to your "x" variable in calculus) to produce an "output set" (like your "y" values). What differentiates a map from a scalar function (e.g. y=mx+b) is that the "x" and "y" here are actually each multi-dimensional co-ordinates rather than a single variable. 1-D functions can be modelled with metal wire, while 2-D functions would need foil surfaces. Conformal maps What makes a conformal map unique is that it is said to preserve angles between any localised points on the input set and their transformed output. Angles between points on a multi-dimensional surfaces are representable as a set of partial derivatives indicating the "slope" between the points in as many dimensions as needed for the dimensionality of the surface. |Wolfram:/Mathworld/Conformal Mapping> "A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation ( w=f(z) ) that preserves local angles. An analytic function is conformal at any point where it has a nonzero derivative. Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping is extremely important in complex analysis, as well as in many areas of physics and engineering. "|Wolfram:/Mathworld/Conformal Mapping> "A mapping that preserves the magnitude of angles, but not their orientation is called an isogonal mapping(Churchill and Brown 1990, p. 241)."|Wolfram:/Mathworld/Conformal Mapping> Conformal Field Theory https://en.wikipedia.org/wiki/Conformal_field_theory "A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications1 to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points." AdS/CFT correspondence The AdS/CFT correspondence is a conjecture proposed by Juan Maldacena in 1997 which states that the boundary of Anti-de Sitter space (AdS) can be regarded as the "spacetime" for a conformal field theoryhttps://en.wikipedia.org/wiki/AdS/CFT_correspondence#The_idea_of_AdS/CFT. This correspondence has been revolutionary in the field of quantum gravity because it shows that cosmological theories that are compatible with Einstein's concepts of gravity can be equivalent to quantum field theories that exists in a spacetime with one less dimensionhttps://en.wikipedia.org/wiki/AdS/CFT_correspondence#Applications_to_quantum_gravity - forming a 'holograph' of the higher dimensional image: hence an example of the broader 'holographic principle'. |Hawking & Hertog:/2018/A smooth exit from eternal inflation?> "The usual theory of inflation breaks down in eternal inflation. We derive a dual description of eternal inflation in terms of a deformed Euclidean CFT located at the threshold of eternal inflation. The partition function gives the amplitude of different geometries of the threshold surface in the no-boundary state. Its local and global behavior in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal to the round three-sphere and essentially zero for surfaces with negative curvature. Based on this we conjecture that the exit from eternal inflation does not produce an infinite fractal-like multiverse, but is finite and reasonably smooth."|Hawking & Hertog:/2018/A smooth exit from eternal inflation?> - Abstract References Category:Field Theory Category:Quantum Cosmology Category:Mathematics Category:Physics